Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T15:10:47.287Z Has data issue: false hasContentIssue false
  • English
  • Français

THE DENSITY OF SUBGROUP INDICES

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 October 2008

ANER SHALEV
ANER SHALEV*
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a group G and a real number x≥1 we let sG(x) denote the number of indices ≤x of subgroups of G. We call the function sG the subgroup density of G, and initiate a study of its asymptotics and its relation to the algebraic structure of G. We also count indices ≤x of maximal subgroups of G, and relate it to symmetric and alternating quotients of G.

Keywords

subgroup indicesspecial linear groups

MSC classification

Secondary: 20E07: Subgroup theorems; subgroup growth 20F69: Asymptotic properties of groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 257 - 267
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Cameron, Peter J., Neumann, Peter M. and Teague, David N., ‘On the degrees of primitive permutation groups’, Math. Z. 180 (1982), 141–149.CrossRefGoogle Scholar
[2]Everitt, Brent, ‘Alternating quotients of Fuchsian groups’, J. Algebra 223 (2000), 457–476.CrossRefGoogle Scholar
[3]Golod, E. S., ‘On nil-algebras and finitely approximable p-groups’, Izv. Akad. Nauk. SSSR Ser. Mat. 28 (1964), 273–276.Google Scholar
[4]Grigorchuk, R. I., ‘On Burnside’s problem on periodic groups’, Funktsional Anal. i Prilozhen 14 (1980), 53–54.CrossRefGoogle Scholar
[5]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 5th edn (Oxford University Press, New York, 1979).Google Scholar
[6]Heath-Brown, D. R., Praeger, Cheryl E. and Shalev, Aner, ‘Permutation groups, simple groups, and sieve methods’, Israel J. Math. 148 (2005), 347–375. (Furstenberg Volume).CrossRefGoogle Scholar
[7]Kassabov, Martin and Nikolov, Nikolay, ‘Cartesian products as profinite completions’, Int. Math. Res. Not., ID 72947 (2006), 17 pp.CrossRefGoogle Scholar
[8]Kleidman, Peter and Liebeck, Martin, The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, 129 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[9]Liebeck, Martin  W. and Saxl, Jan, ‘On the orders of maximal subgroups of the finite simple exceptional groups of Lie type’, Proc. London Math. Soc. 55 (1987), 299–330.CrossRefGoogle Scholar
[10]Liebeck, Martin W. and Shalev, Aner, ‘Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky’, J. Algebra 184 (1996), 31–57.CrossRefGoogle Scholar
[11]Liebeck, Martin W. and Shalev, Aner, ‘Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks’, J. Algebra 276 (2004), 552–601.CrossRefGoogle Scholar
[12]Liebeck, Martin W., Pyber, Laszlo and Shalev, Aner, ‘On a conjecture of G.E. Wall’, J. Algebra 317 (2007), 184–197.CrossRefGoogle Scholar
[13]Lubotzky, Alexander and Segal, Dan, Subgroup Growth, Progress in Mathematics, 212 (Birkhäuser, Basel, 2003).CrossRefGoogle Scholar
[14]Nikolov, Nikolay, Strong approximation methods in group theory — LMS/EPSRC short course, (2007). arxiv:math.GR 0803.4165v2.Google Scholar
[15]Shalev, Aner, The density of subgroup indices, II, in preparation.Google Scholar